First-order Modal Languages for the Specification of Multi-agent Systems

Transcription

1 First-order Modal Languages for the Specification of Multi-agent Systems Joint work with Alessio Lomuscio Department of Computing Imperial College London November 24, 2011

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3 Background Propositional modal languages (temporal, epistemic, deontic, coalition logic) are a success story on the applications of logic to computer science. Today a number of tools and techniques based on (some of) these formalisms are available for the specification and verification of distributed and multi-agent systems. NuSMV McMAS Prism SPIN UPAAL etc... Still, as increasing complex scenarios (web services, e-commerce, ACSI project) are tackled by the verification community, the demand for ever more expressive languages is becoming more pressing.

4 The Equipment Purchase Scenario 4. the Buyer submits the POs to the Suppliers 2. the Requester submits the RO to the Buyer Requester 1. the Requester creates a Requisition Order Buyer 3. the Buyer prepares a Procurement Order for each item in the RO 6. either the Buyer is notified or items are shipped Suppliers 5. the Suppliers either accept or reject the POs

5 Extending to First-order First-order logic is typically considered the most convenient setting to formalise natural language: each of the Requisition Order, Procurement Order can be formalised as a database, i.e., a first-order structure; we are interested in model checking formulas like AG id, itm, p(ro(id, itm, p, closed) s K Buyer PO(p, s, itm, accepted)) a Requisition Order can be in state closed only if the Buyer knows that the corresponding Procurement Order has been accepted by some Supplier.

6 Extending to First-order However, first-order (modal) logic is known to be highly undecidable: [HWZ00] Let F be either { N, < } or { Z, < }. Then the set T L 2 T L mo TL(F) is not recursively enumerable. [HWZ00] Let F be one of the following classes of temporal frames: { N, < }, { Z, < }, the class of all strict linear orders. Then T L 2 T L mo TL fin (F) is not recursively enumerable. [WZ01] The two variable monadic fragment of QK (no matter with constants or expanding domains) is not recursively enumerable. [W00] For any first-order common knowledge logic L: The fragment of L with monadic predicates only (without equality and function symbols) is not recursively enumerable. The fragment of L with local constant symbols and the equality symbol only (without predicates and proper function symbols) is not recursively enumerable.

7 The Monodic Fragment Definition (Monodic fragment) Let L be a first-order modal language, the monodic fragment L 1 of L is the set of formulas φ L such that any subformula of φ of the form ψ, for some modal operator, contains at most one free variable. Examples: agent i knows that every resource is universally available until it is requested, y(resource(y) K i ( zavailable(y, z)u xrequest(x, y)) for every process agent i knows that it will eventually try to access every resource, xk i (Process(x) y(resource(y) F Access(x, y)) 6 The monodic fragment contains all de dicto formulas, so the limitation is only on de re formulas.

8 The Monodic Fragment [HWZ00] Let T L T L 1 and suppose there is an algorithm which is capable of deciding, for any T L -sentence φ, whether an arbitrary state candidate for φ is realizable. Let F be one of the following classes of flows of time: { N, < }, { Z, < }, { Q, < }, the class of all finite strict linear orders, any first-order-definable class of strict linear orders. Then the satisfability problem for the T L -sentences in F, and so the decision problem for the fragment TL(F) T L, is decidable. [WZ01] For L = {QK, QK, QT, QK4, QS4} the fragments L ML 2 1, L ML m 1, and L MGF 1 are decidable. [SWZ00] Let L be any of K C n, T C n, KD C n, K4 C n, S4 C n, KD45 C n, S5 C n. Then the following fragments are decidable: the monodic fragment, the two-variable fragment, the guarded fragment. However, Let L be any of the logics above. Then the fragment L 1 in the language QCL = 1 is not recursively enumerable.

9 Axiomatizing the Monodic Fragment In this talk we show that the axiomatisations for temporal epistemic logic available at the propositional level can be lifted to the monodic fragment. This work builds on previous contributions: - on the monodic fragment of first-order temporal [WZ02] and epistemic logic [SWZ00, SWZ02]; - on propositional temporal epistemic logic [HvdMV03]. We consider well-known properties of multi-agent systems including perfect recall, synchronicity, no learning, and having a unique initial state.

10 First-order Temporal Epistemic Languages Let A = {1,..., i,..., m} be a set of agents. Definition (terms and formulas in L m ) t ::= x c φ ::= P k (t 1,..., t k ) ψ ψ ψ xψ ψ ψuψ K i ψ The language L m combines at first order: the LTL operators next and until U, and the epistemic operators K i for each agent i A. The language LC m extends L m with the common knowledge operator C.

11 Quantified Interpreted Systems Interpreted systems are the typical formalism for reasoning about knowledge in multi-agent systems [FHMV95]. Let S L e L 1... L m be the set of global states, where L i is the set of local states for each agent i in A and for the environment e. Definition (QIS) A quantified interpreted system is a tuple P = R, D, I such that - R is a non-empty set of runs r : N S - D is a non-empty set of individuals - I is a first-order interpretation of individual constants and predicate symbols. QIS m is the class of all quantified interpreted systems P with m agents.

12 Quantified Interpreted Systems Consider a formula φ L m (resp. LC m), a point (r, n) P, and an assignment σ : Var D: (P σ, r, n) = P k (t 1,..., t k ) iff I σ (t 1 ),..., I σ (t k ) I (P k, r, n) (P σ, r, n) = ψ iff (P σ, r, n) = ψ (P σ, r, n) = ψ ψ iff (P σ, r, n) = ψ or (P σ, r, n) = ψ (P σ, r, n) = xψ iff for all a D, (P σx a, r, n) = ψ (P σ, r, n) = ψ iff (P σ, r, n + 1) = ψ (P σ, r, n) = ψuψ iff there is n n such that (P σ, r, n ) = ψ and for all n n < n, (P σ, r, n ) = ψ (P σ, r, n) = K i ψ iff r i (n ) = r i (n) implies (P σ, r, n ) = ψ (P σ, r, n) = Cψ iff for all k N, (P σ, r, n) = E k ψ where E 0 ψ = V i A K iψ = Eψ, and E k+1 ψ = EE k ψ. The definitions of truth and validity in a class of QIS are standard.

13 Quantified Interpreted Systems A QIS P satisfies synchronicity iff time is part of the local state of each agent, i.e., r i (n) = r i (n ) implies n = n perfect recall iff each agent remembers everything that has happened to her during the run (i.e., local states are histories) no learning iff no agent aquires new knowledge during the run unique initial state iff all runs begin from the same global state, i.e., for all r, r R, r(0) = r (0) The superscripts sync, pr, nl, uis denote specific subclasses of QIS m. For instance, QIS pr,sync m contains the synchronous QIS with perfect recall.

14 The Systems QKT m The system QKT m combines at first order: the linear temporal logic LTL, and the multi-modal epistemic logic S5 m. Classical first-order logic Axioms for Time classical propositional tautologies φ ψ, φ ψ xφ φ[x/t] φ ψ[x/t] φ xψ, x not free in φ (φ ψ) ( φ ψ) φ φ φuψ ψ (φ (φuψ)) φ φ χ ψ χ χ (φuψ) Axioms for Knowledge K i (φ ψ) (K i φ K i ψ) K i φ φ K i φ K i K i φ K i φ K i K i φ φ K i φ Barcan formulas xφ x φ K i xφ xk i φ

15 More Systems Consider the following axioms for common knowledge: C1 C2 Cφ (φ ECφ) φ (ψ Eφ) = φ Cψ The system QKTC m extends QKT m with C1 and C2. Consider the following axioms on the interaction between time and knowledge: KT1 K i φ K i φ KT2 K i φ (K i ψ K i χ) K i ((K i φ)u((k i ψ)u χ)) KT3 (K i φ)uk i ψ K i ((K i φ)uk i ψ) KT4 K i φ K i φ KT5 K i φ K j φ We use 1,..., 5 as superscripts to denote the systems obtained by adding to QKT m (resp. QKTC m) any combination of KT1-KT5. For instance, the system QKT 2,3 m extends QKT m with KT2 and KT3.

16 Soundness Results A QIS P satisfies any of KT1-KT5 if P satisfies the corresponding condition: Axiom KT1 KT2 KT3 KT4 KT5 Condition on QIS perfect recall, synchronicity perfect recall no learning no learning, synchronicity all agents share the same knowledge

17 Completeness Results Theorem (Completeness) The systems in the first and second column are sound and complete w.r.t. the corresponding classes of QIS in the third column. Systems QKT m QKTC m QIS m, QIS sync m QKTm 1 QKTm 2 QKTm 3 QKTm 4 QKTm 2,3 QKT 1,4 m QIS pr m, QIS pr,uis m QIS pr,sync m QIS nl m QIS nl,sync m QIS nl,pr m QIS nl,pr,sync m QIS, QIS uis m, QIS sync,uis m, QIS pr,sync,uis m QKT 2,3 1 QIS nl,uis 1, QIS nl,pr,uis 1 QKTm 1,4,5 QKTCm 1,4,5 QIS nl,sync,uis m, QIS nl,pr,sync,uis m

18 Completeness Results All completeness results available at the propositional level [HvdMV03] can be lifted to the monodic fragment of L m (resp. LC m). In the general case there is no complete axiomatisation of the validities on QIS nl,uis m and QIS nl,pr,uis m. This is already the case at the propositional level. However, for m = 1, QIS nl,pr,uis 1 = QIS nl,uis 1 = QIS nl,pr 1. Hence QKT 2,3 1 is a complete axiomatisation for QIS nl,uis 1 and QIS nl,pr,uis 1.

19 Idea of the Completeness Proof Two ingredients are needed in order to prove completeness: Lemma quasimodels, introduced in [HWZ00]; monodic-friendly Kripke models, a Kripke-style semantics for L 1 m (resp. LC 1 m). For every monodic formula φ, satisfiability in mf-kripke models satisfiability in QIS. satisfiability in quasimodels satisfiability in mf-kripke models. Synchronicity, perfect recall, no learning, and unique initial state are preserved. Lemma a monodic φ is consistent w.r.t. a system S φ is satisfiable in a quasimodel for S. Theorem As a result, The system S is complete for the corresponding class of QIS.

20 Monodic-friendly Kripke Models Definition (mf-kripke model) A monodic-friendly Kripke model is a tuple M = R, { i } i A,a D, D, I where - R is a non-empty set of indexes r, r,... - D is a non-empty set of individuals - I is a first-order interpretation of individual constants and predicate symbols - for i A, a D, i,a is an equivalence relation on the points (r, n), for r R and n N (M σ, r, n) = K i ψ[y] iff (r, n) i,σ(y) (r, n ) implies (M σ, r, n ) = ψ (M σ, r, n) = Cψ[y] iff for all k N, (M σ, r, n) = E k ψ[y] The class of mf-kripke models with m agents is denoted by K m. We can consider mf-kripke models satisfying synchronicity, perfect recall, no learning, or having a unique initial state. mf-kripke models do not necessarily satisfy BF and axiom K!!

21 Monodic-friendly Kripke models Monodic-friendly Kripke models are necessary to prove completeness for systems encompassing either perfect recall or no learning. Lemma We are only able to prove that For every monodic formula φ, satisfiability in quasimodels satisfiability in mf-kripke models. Lemma not in general Kripke models. However, Let M be a mf-kripke model satisfying BF and axiom K. There exists a map g : K m QIS m such that for every φ, M satisfies φ g(m) satisfies φ The map g preserves any of synchronicity, perfect recall, no learning, or having a unique initial state.

22 Quasimodels A quasimodel for a monodic formula φ L 1 m (resp. LC 1 m) is a relational structure whose points are sets of sets of subformulas of φ. Quasimodels have been introduced in [HWZ00], and then applied to first-order temporal [WZ02] and epistemic logic [SWZ00, SWZ02]. Here we combine quasimodels with the techniques in [HvdMV03].

23 Quasimodels Fix a monodic φ in L 1 m (resp. LC 1 m), sub xφ is the set of subformulas of φ containing at most the free variable x. Also, sub xφ is closed under negation. For k N we define the closures cl k φ and cl k,i φ by mutual recursion. Definition Let cl 0φ = sub xφ. For k 0, cl k+1 φ = S i Ag cl k,iφ. For k 0, i Ag, cl k,i φ = cl k φ {K i (ψ 1... ψ n), K i (ψ 1... ψ n) ψ 1,..., ψ n cl k φ}. Let ι be any finite sequence i 1,..., i k of agents such that i n i n+1. If ι is the empty sequence ɛ then cl ιφ = cl ad(φ) φ. If ι = ι i, then cl ιφ = cl k,i φ for k = ad(φ) ι. Definition (Type) A ι-type t is any boolean saturated subset of cl ιφ.

24 Quasimodels Types represent individuals in a particular point. Thus, points are represented by collections of types, i.e., state candidates. To each state candidate C is associated a formula α C L 1 m. Now we need a way of identifying the same individual across points. Definition (Object) Let a frame F be a tuple R, { i,a } i A,a D, D where R, i,a and D are defined as for mf-kripke models, and whose points are state candidates. For a D, an object in F is a map ρ a associating with every (r, n) F a type ρ a(r, n) C r,n such that: 1 ρ a(r, n) ρ a(r, n + 1) is consistent 2 if (r, n) i,a (r, n ) then ρ a(r, n) K i ρ a(r, n ) is consistent 3 χuψ ρ a(r, n) iff there is n n such that ψ ρ a(r, n ) and χ ρ a(r, n ) for all n n < n 4 if K i ψ ρ a(r, n) then there is some (r, n ) such that (r, n) i,a (r, n ) and ψ ρ a(r, n ).

25 Quasimodels We can now provide the definition of quasimodels. Definition (Quasimodel) A quasimodel for φ is a tuple Q = R, { i,ρ } i A,ρ O, O such that R, { i,ρ } i A,ρ O, O is a frame, and 1 φ t for some t C r,n 2 α Cr,n α Cr,n+1 is consistent 3 if (r, n) i,ρ (r, n ) then ρ(r, n) K i ρ(r, n ) is consistent 4 for all t C r,n there exists an object ρ O such that ρ(r, n) = t 5 for all c conφ, the function ρ c such that ρ c (r, n) = t c C r,n is an object in O.

26 Quasimodels Lemma For every monodic formula φ, satisfiability in quasimodels satisfiability in QIS. It s only left to prove that Lemma a monodic φ is consistent w.r.t. a system S φ is satisfiable in a quasimodel for S. The proof of this lemma depends on the particular system S.

27 Conclusions and Future Work In this talk: we presented a number of classes of QIS satisfying perfect recall, synchronicity, no learning, and having a unique initial state. we proved that the axiomatisations available at the propositional level can be lifted to the monodic fragment of L m (resp. LC m). In future work we aim to: analyze CTL and epistemic modalities interpreted on QIS. explore the issues pertaining to the decidability of the logics here discussed.

28 Conclusions Thank you!

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